Recursive filtering is a form of filtering where the output of the filter is fed back as an input. This can achieve a long impulse response for a tiny computational effort. It is particularly useful in computing a running mean. A running mean could be implemented as a low-pass filter in the frequency domain, but it is generally much better to avoid transform space. Physical space is cheaper, it allows for variable coefficients, and it permits a more flexible treatment of boundaries. Geophysical datasets are rarely stationary over long distances in either time or space, so recursive filtering is particularly helpful in statistical estimation.
The purpose of most filters is to make possible the observation of important weak events that are obscured by strong events. One -dimensional filters can do this only by the selection or rejection of frequency components. In two dimensions, a different criterion is possible, namely, selection by dip.
Dip filtering is a process of long-standing interest in geophysics (Embree, Burg, and Backus ). Steep dips are often ground-roll noise. Horizontal dips can also be noise. For example, weak fault diffractions carry valuable information, but they may often be invisible because of the dominating presence of flat layers.
To do an ordinary dip-filtering operation (``pie slice''), you simply transform data into -space, multiply by any desired function of , and transform back. Pie-slice filters thus offer complete control over the filter response in dip space. While the recursive dip filters are not controlled so easily, they do meet the same general needs as pie-slice filters and offer the additional advantages of
The causality property offers an interesting opportunity during data recording. Water-velocity rejection filters could be built into the recording apparatus of a modern high-density marine cable.